Project
Informative prior specification for nonlinear Gaussian processes

Many real-world phenomena cannot be adequately described by linear relationships, including those in medical, natural and social sciences. In psychology, nonlinearity naturally appears in longitudinal data that capture temporal variations and in data reflecting complex social interactions and dynamical psychological states (Preacher et al., 2006; Simonsohn, 2021; Chow, 2013; Asselman et al., 2020). To model and study such complex, nonlinear processes, conventional statistical tools are often insufficient. Instead, flexible and interpretable nonlinear modeling frameworks are required.

One powerful approach is the use of Bayesian Gaussian processes (GPs) (O’Hagan, 1978; Neal, 1999; Rasmussen & Williams, 2006; Vanhatalo et al., 2013; Cheng et al., 2019). They offer a nonparametric framework for modeling unknown functions without specifying a fixed functional form. A key advantage of this approach lies in the ability to incorporate prior knowledge about the underlying process in an explicit and mathematically precise manner and quantify its concordance with the empirical data.

In most Bayesian models, the specification of priors is a central aspect of the analysis, and extensive research has focused on the construction of informative and non-informative priors that encode prior knowledge with a desired level of precision and influence on the final estimation (Berger, 1990; Ibrahim & Chen, 2000). The specificity of such priors can range from the general shape and constraints to the actual distribution characteristics.

However, in Gaussian process modeling, priors have been developed in a different direction. GP prior defines a distribution over functions and is characterized by a mean function and a covariance (kernel) function. The mean function describes the expected function values, while the kernel specifies properties such as smoothness, periodicity, and correlation length through the choice of kernel family and its hyperparameters. Although flexible, this formulation rarely encodes semantically meaningful information such as monotonicity, value limits and other function characteristics that can be expected from prior theoretical knowledge and common sense. For instance, when modeling the process of work integration for a new employee, one could reasonably expect that the trajectory starts at zero, increases monotonously, and eventually reaches a plateau, and, ideally, this information should be explicitly included in the prior.

The aim of this project is to develop a new generation of safe, informative Gaussian process priors that extend beyond conventional mean and covariance function specifications. These priors will be designed to incorporate domain-specific constraints such as monotonicity, bounded value ranges, and expected function shapes derived from theoretical or empirical knowledge.

Another goal of this project is to analyze the meaning and quantification of prior “informativeness”, specifically, how the strength of the prior influences posterior inference, and to develop methods for moderating this influence based on the level of certainty or relevance of the prior knowledge. Finally, the project will enrich Gaussian process analysis tools by incorporating priors from different kernel families and mechanisms for integrating historical or training data into prior specification.

Collectively, these developments aim to enhance the expressiveness, interpretability, and practical relevance of Gaussian process modeling, expanding the possibilities for meaningful exploration of nonlinear phenomena in psychology and beyond.

Supervisors
Prof. Dr. Ir. Joris Mulder

Financed by
ERC Consolidator Grant 2022

Period
1 September 2025 – 1 September 2029